Performance of Probability Distributions and Plotting Positions in

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O.B. Adeboye and M.O. Alatise and “Performance of Probability Distributions and Plotting Positions in Estimating the Flood of River Osun at Apoje Sub-basin, Nigeria”. Agricultural Engineering International: the CIGR Ejournal. Manuscript LW 07 007. Vol. IX. July, 2007.

Performance of Probability Distributions and Plotting Positions in Estimating the Flood of River Osun at Apoje Sub-basin, Nigeria

Omotayo B. Adeboye* and Michael O. Alatise

Department of Agricultural Engineering, Federal University of Technology, P.M.B 704, Akure, Ondo State, Nigeria; +234(0)8025306602; Email:*[email protected]

ABSTRACT

The occurrence of flood and its attendant damage to both life and property in recent times especially in rural and urban dwellings and also in agricultural establishments have generated a lot of concern in Nigeria. The search for solution to problems on flood damage and risk which can probably be avoided if adequate and precise flood forecasting mechanisms are put in place has provoked this study. For the purpose of statistical analysis, 18-year streamflow record of River Osun at Apoje gauging station was obtained from Ogun-Osun River Basin and Rural Development Authourity Abeokuta, Nigeria. The annual hydrograph was drawn for the station by plotting the peak discharges against their hydrologic years. The peak discharges were fitted to the three major statistical distributions namely normal, log-normal and log-Pearson type (iii) while seven plotting positions of Hazen, Weibull, Blom, Cunnane, California, Gringorton and Chegodajev were used in determining their probabilities of exceedance. Results obtained showed that the annual maximum discharges at Apoje station varied from 169m3/s to 400m3/s within duration of 18 years. Weibull’s plotting position combined with normal distribution gave the highest fit, most reliable and accurate predictions of the flood in the study area having the coefficient of determination R2 and root mean square error RMSE of 0.9950 and 35.09 m3/s respectively. Keywords: Apoje, flood, plotting positions, statistical distributions, Ogun-Osun river basin

annual maximum discharges.

1. INTRODUCTION

Water is the most vital requirement for human survival and it has been described as the elixir of life and a major agent that ensures continuity in an ecosystem (Tadesse, 2006). Scarcity of water and the presence of it in abundance especially if not controlled have their consequences in every environment. Hydrologist finds it difficult to make accurate prediction of flood estimates using limited historic information of runoff, rainfall, river stages. These can be attributed to lack of trained personnel and equipment for adequate assessment of these quantities on systematic basis. Where these data are available, they are characterised by gaps, which in most cases limit the inferences from their statistical analysis and eventually lower their usefulness for planning purposes (Viessman and Lewis, 1996). Using mathematical equations and statistical models, these gaps can be filled or the series extended to a longer

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period of time (Claps and Laio, 2003). In stochastic hydrology, the common and result oriented probability distributions in use are the normal, lognormal and the log Pearson type (iii), gamma, Weibull’ and Gumbell (Hromadka and Whitley, 1989; Moughamian, et al, 1987; Robert, 1987). The normal and the lognormal distributions fit adequately to the peak rainfall and streamflow while for the extreme hydrologic variables, the Weibull and Gumbell distributions are used (Aksoy, 2000). The uses of direct statistical and regional techniques in flood frequency estimation have long been advocated for, but have witnessed little attention. The regional methods allow the design event to be estimated at ungauged site and the reliability of the estimated events increases due to the increase in the incorporation of additional information (Hoskins and Wallis, 1997). However, the reliability of the predicted events for the design purposes at ungauged sites is significantly lower when compared with the values at a gauged site (Kjeldsen et al, 2001). Hebson and Cunnane (1987) showed that the values of the mean annual floods (MAF) at ungauged sites are less precise than the estimates at gauged site even with only one year of reliable data. For the purpose of efficiency, most hydrologists prefer the Maximum Likelihood, and Bayesian methods (Hirsch and Stedinger, 1987). Most engineering decisions are made with the use of graphic display, but for the purpose of extrapolation of the curves, the probability curves are used.

The subject of plotting positions is not new to hydrologists and statisticians. Plotting positions have been used in estimating magnitudes of hydrological events and their corresponding return periods, detecting outliers, fitting distributions to the data and in evaluating the adequacy of fit of the alternative parametric floods frequency models. A variety of plotting position formulae have been proposed in hydrological and statistical literature during the past 50 years. The optimum formulae are selected based on the purpose of investigation and the distribution under consideration (Van-Thanh-Van Nguyen, et al, 1989).

At present, there is no single universally accepted model, rather, a whole group of model such as the Gumbel, the Log-Logistic, the lognormal, and the log-Pearson type (iii) distributions have been suggested for predicting the magnitudes of such extreme events (Van-Thanh-Van Nguyen, et al, 1989). Therefore the question of better fit among numerous models is always welcome (Topaloglue, 2002). The performances of models have been assessed using statistical tools such as root mean square error (RMSE) and coefficient of determination (Van Bladeren, 1993 and Ekanayake and Cruise, 1993). The objective of this study was to determine the flood characteristics of the study area by using three major statistical methods and seven (7) different plotting positions.

2. MATERIALS AND METHODS

2.1 The Study Area The Osun River Basin is located in an area whose boundaries are approximately latitudes 8020’N and 6030’N and longitudes 5010’E and 3025’E (Fig.1). The area of the basin is approximately 16,700Km2 (FRN, 1982). Generally, the Osun basin’s climate is influenced by the movement of the inter-tropical convergence Zone (ITCZ), a quasi-stationary boundary

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zone which separates the sub-tropical continental air mass over the Sahara and the equatorial maritime air mass over the Atlantic Ocean as reported by Adeboye, (2005). In the wet season, the mean rainfall ranges between 1,020 and 1,520 mm in the South of the basin, but in the North, it is less than 1,020 mm. In the North and South, the mean dry season rainfall varies from 127 to 178 mm and 178 to 254 mm respectively. Water resources in Osun basin include the surface water and groundwater. Surface water plays the prominent role in the basin but the limited stream flow records create problems in water resources assessment. Generation of long record is done by streamflow synthesis, of rainfall records. Apoje sub basin is selected for this study because it is an agrarian community which serve as the food basket for a major part of the south-western Nigeria.

Figure 1: Hydrological networks of Ogun-Osun River Basins.

The Methods of Data Analysis

Daily steamflow data for the duration of 18-years (1982-1999) at Apoje gauging station were obtained from Ogun-Osun River Basin and Rural Development Authority, Abeokuta, Nigeria for the purpose of this study. The annual peak discharges were selected and ranked in descending order of magnitudes based on the recommendations of the (USWRC, 1981). The peak discharges were plotted against their hydrologic years (1982-1999) in order to determine the spatial variations of the annual maximum discharges in the sub-basin. The mean of the

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annual flows were also plotted against the water years. The probabilities of exceedence of the discharges were determined using the seven plotting positions shown in Table 1.

Table 1: Plotting positions used in determining the flood probabilities of exceedence

Plotting Positions

Formulae

Hazen (1914),

n

mQQP T5.0)( −

=≥

Weibull (1939), =≥ )( TQQP

1+nm

Blom (1958), =≥ )( TQQP

25.0375.0

+−

nm

Cunnane (1978), =≥ )( TQQP

2.04.0

+−

nm

California (1923), =≥ )( TQQP

nm

Gringorton(1963), =≥ )( TQQP12.044.0

+−

nm

Chegodajev(1955) =≥ )( TQQP

4.03.0

+−

nm

From the Table 1 Q is anticipated streamflow, (m3/s),

TQ is streamflow of estimated return period to be equalled or exceeded, (m3/s), m is rank, and n is number of observations. The return periods of the anticipated discharges were determined by finding the reciprocals of the exceedence probabilities and is expressed by

pTr

1= (1)

where rT is return period;

p is probability of exceedence that is, the probability that a given flood is equalled or exceeded.

Three distribution models were considered for the purpose of accurately estimating the flood magnitudes and frequencies of River Osun at Apoje gauging station, Nigeria.

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3. FREQUENCY ANALYSIS METHODS

3.1 Normal Distribution For a symmetrically distributed data, the most appropriate distribution of continuous variable is the normal distribution which is also called the Gaussian distribution (Tilahun, 2006). The probability density function (PDF) of this distribution model according to (Chow et al, 1988) is given by

22

21 )(

zzf −= l

π (2)

where z is standard normal variable, and e is exponential.

The statistical parameters, mean and standard deviation of the annual maximum discharges were determined using the method of moment and their respective equations are expressed as

∑=

=n

iQ

nQ

1max

1 (3)

( )

2

1max

1−

−=∑=

n

QQS

n

iQ (4)

Where Q is mean of the annual maximum discharges (m3/s),

maxQ is annual maximum discharges (m3/s),

QS is standard deviation of the annual maximum discharges (m3/s), and n is as previously defined.

In this distribution, the intermediate variable was determined using the expression

⎟⎟⎠

⎞⎜⎜⎝

⎛=

pw 1ln 5.00 ≤< p (5)

where w is intermediate variable, and p is as previously defined.

The frequency factors corresponding to the return periods of the ranked annual maximum discharges were determined using the expression

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z = 32

2

001388.0189269.0432788.11010328.0802853.0515517.2

wwwwww

+++++

− (6)

where, p is probability of exceedence, w is intermediate variable, z is standard normal variable or the frequency factor.

The predicted flood at various return periods were determined using the mathematical expression QT SzQQ ⋅+= (7)

TQ , Q , z , and QS are as previously defined.

3.2 Lognormal Distribution Large numbers of hydrological continuous variable random variables tends to be asymmetrically distributed. It is advantageous to transform the distribution to a normal distribution by taking the logarithms of the annual maximum discharges (Tilahun, 2006; USWRC, 1981). The probability density function (PDF) under this distribution is given as

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛ −−= 2

y

2

exp2

1

y

y

xxf

σμ

πσ (8)

In this flood analysis, the logarithms of the annual maximum discharges were taken to base 10. Using the method of moment, the mean and the standard deviation of the ranked annual maximum discharges were determined by the expression:

∑=

=n

iQ

ny

1maxlog1 (9)

( )( )1

1

2

−

−=∑=

n

yyS

n

iy (10)

respectively, where y is mean of y (m3/s),

yS is standard deviation (m3/s),

maxlogQy = (m3/s), and n is as previously defined,

The intermediate variables and frequency factors corresponding to the ranked annual maximum discharges were determined by Equations (5) and (6) respectively. The statistical variate and the predicted discharges under this distribution were determined respectively by the expressions

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yT Szyy ⋅+= (11)

( )ySzyTQ ⋅+=10 (12)

Where Ty is variate of the annual maximum discharges at return periods T (years) ,

y is mean of the logarithm of the historic annual maximum discharges (m3/s),

yS , TQ and z are as previously defined.

3.3 Log Pearson Type (iii) Distribution This is referred to as the three parameter fit. Due to its performance in stochastic hydrology, it has been adopted in some countries as the standard distribution for flood frequency analysis (Sumioka et al, 1997). The probability model is given as

)()()(

)91

βελ ελββ

Γ−

=−−−

xyxf

yl (13)

where,

,log xy = β

λ yS= ,

2

)(2

⎥⎦

⎤⎢⎣

⎡=

yCs

β , βε ySy == .

In addition to the mean and the standard deviation of the historic annual maximum discharges determined using Equations (9) and (10), the third parameter, the skew coefficient, was determined using the expression.

( )( )( )( )321

1

3

−−−

−=

∑=

nnn

yynC

n

is (14)

Where sC is the coefficient of skewness; and n , y and y are as previously defined.

Using the Log Pearson type (iii) distribution, the frequency factors corresponding to the predicted annual maximum discharges was given by (Kite, 1977) as

( ) ( ) ( ) 5432232

31163

11 kzkkzkzzkzzKT ++−−−+−+= (15)

where TK is frequency factor,

k is expressed as 6sC , and

z is as previously defined. At each return period, the predicted discharges were determined by

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( )yT SKyTQ ⋅+=10 (16)

In order to compare a model output to the observed data, criteria for making such a comparison must be identified as suggested by Green and Stephenson (1985). Visual comparison of the plotted predicted and observed discharges can be very useful in assessing the accuracy of the model output. However, additional statistical analysis is needed because visual comparison usually tends to be very subjective (Tewolde and Smithers, 2006). Three statistical procedures were used in these analyses for evaluating the performance of the distributions and these are coefficient of determination, absolute differences between the predicted and observed discharges and root mean square error (RMSE). The RMSE between the predicted and observed discharges were determined using the equation given by (O’Donnell, 1995) and is expressed as

( )5.0

1

21⎥⎦

⎤⎢⎣

⎡−= ∑

=

−n

i

OPnRMSE (17)

Where RMSE is root mean square error (m3/s), P is predicted discharges under each distribution (m3/s), Q is observed discharges(m3/s), and n is as previously defined.

4. RESULTS AND DISCUSSION 4.1 Hydrograph The time series of the annual hydrograph is shown in Fig.2 The highest discharge of 400m3/s was observed in 1987 and declined to 169m3/s in the year 1999.The minimum peak flow in the Apoje sub-basin was 72m3/s. The minimum and maximum Mean Annual Flows (MAF) were 82 and 189m3/s respectively. The difference in magnitudes can be attributed to diversion of water to the control structures at Asejire and Ede which are along the course of River Osun. According to Kottegoda, (1980), emphasis in time series analyses should be laid on the mechanisms that generate the data but not necessarily on the future sequence of events over a period of time. 4.2 Flood Frequency Analysis The annual maximum discharges were fitted to the normal, lognormal and log Pearson type iii distributions and the exceedence probabilities determined using the various equations in Table 1. The determined probabilities of exceedence of the peak discharges before subjecting them to the statistical distributions are shown in Table 2.

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0

80

160

240

320

400

480

1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000

Hydrologic years

Ann

ual f

low

s (m

3 /s)

Annual peak discharges Mean of annual flow

Figure 2 Annual flows of River Osun at Apoje gauging station from 1982 to 1999.

Table 2: Probabilities of exceedence of the annual maximum discharges using seven plotting positions

Years Annual maximum discharges

(m3/s)

Hazen

Weibull

Blom

Cunnane

California

Gringorton

Chegodajev

1982 306 0.0278 0.0526 0.0342 0.0329 0.0556 0.0309 0.0380 1983 200 0.0833 0.1053 0.0890 0.0879 0.1111 0.0861 0.0924 1984 255 0.0833 0.1053 0.0890 0.0879 0.1111 0.0861 0.0924 1985 372 0.0833 0.1053 0.0890 0.0879 0.1111 0.0861 0.0924 1986 236 0.1389 0.1579 0.1438 0.1429 0.1667 0.1413 0.1467 1987 400 0.1944 0.2105 0.1986 0.1978 0.2222 0.1965 0.2011 1988 363 0.2500 0.2632 0.2534 0.2528 0.2778 0.2517 0.2554 1989 360 0.3056 0.3158 0.3082 0.3077 0.3333 0.3068 0.3098 1990 337 0.3611 0.3684 0.3630 0.3626 0.3889 0.3620 0.3641 1991 372 0.4167 0.4211 0.4178 0.4176 0.4444 0.4172 0.4185 1992 372 0.4722 0.4737 0.4726 0.4725 0.5000 0.4724 0.4728 1993 273 0.5278 0.5263 0.5274 0.5275 0.5560 0.5278 0.5272 1994 209 0.5833 0.5789 0.5822 0.5824 0.5611 0.5828 0.5815 1995 354 0.6389 0.6616 0.6370 0.6374 0.6667 0.6379 0.6359 1996 299 0.6944 0.6842 0.6918 0.6923 0.7222 0.6932 0.6900 1997 227 0.7500 0.7368 0.7466 0.7423 0.7778 0.7483 0.7446 1998 169 0.8056 0.7895 0.8016 0.8022 0.8333 0.8035 0.7989 1999 195 0.9167 0.8421 0.8562 0.8571 0.8889 0.8587 0.8533

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4.3 Hazen’s Plotting Position The observed discharges in Figs.3a, b and c at return periods of 25, 50 and 100 years were 404, 412 and 417m3/s respectively and the coefficient of determination R2 was 0.9761. Figure 3a shows the graphical illustration of the annual maximum discharges against the non-exceedence probability for the normal distribution using the Hazen’s plotting position. From this figure, it can be seen that at return periods of 25, 50 and 100 years, the predicted discharges were 410, 416 and 420m3/s respectively. The coefficient of determination R2 was 0.9908. These estimates are higher and compares well with observed discharges. Minimum absolute differences between the predicted and the observed discharges were obtained under this distribution. The root mean square error (RMSE) was 35.84m3/s (see Table 3). Figure 3b shows similar graphical illustration as in Fig.3a but under the lognormal distribution. At return periods of 25, 50 and 100 years, the predicted discharges were 434, 441 and 445m3/s respectively and R2 was 0.9752. These flood estimates are higher and do not compare well with the observed discharges. The RMSE was 43.09m3/s, the highest under this plotting position. The highest absolute differences between the observed and predicted discharges were obtained under this distribution. Figure 3c gives the same graphical illustration as Figs.3a and 3b but this time under the log Pearson type (iii) distribution. At return periods of 25, 50, and 100 years, the predicted discharges were 427, 435 and 438m3/s respectively and R2 was 0.9890. The normal distribution gave the highest fit using this plotting position. 4.4 Weibull’s Plotting Position The observed discharges in Figs 4a, b and c at return periods of 25, 50 and 100 years were 417, 426 and 431m3/s respectively and coefficient of determination R2 was 0.9761. Figure 4a shows the graphical illustration of the annual maximum discharges against the probabilities of non-exceedence for the normal distribution using Weibull’s plotting position. From the figure, at return periods of 25, 50 and 100 years, the predicted discharges were 408, 414 and 417m3/s respectively and R2 for the predicted discharges was 0.9950. The flood estimates compared well with the observed discharges having root mean square error (RMSE) of 35.09m3/s which was the smallest under this plotting position (see Table 3). Similar graph of lognormal distribution is shown in Fig.4b. At return periods of 25, 50 and 100 years, the predicted discharges were 428, 435 and 439m3/s respectively and R2 for the predicted discharges was 0.9860. Minimum absolute differences between the observed and predicted discharges were obtained under this distribution. Figure 4c shows the graphical illustration for log-Pearson type (iii) distribution. At return periods of 25, 50 and 100 years, the predicted discharges 433, 440 and 444m3/s respectively and R2 values was 0.9942. By comparing the three statistical distributions, it is evident that the normal distribution gave the highest degree of correlation for the predicted discharges. 4.5 Blom’s Plotting Position The observed discharges in Figs 5a, b and c were 409, 422 and 427m3/s at return periods of 25, 50 and 100 years respectively. The graph of annual maximum discharges against the non-exceedence probabilities for the normal distribution using Blom’s plotting position is shown

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in Fig.5a. At the return periods of 25, 50 and 100 years, the predicted discharges were 413, 419 and 422m3/s respectively and R2 was 0.9924. The root mean square error (RMSE) was 35.75m3/s (see Table 3). These predicted discharges compares well with the observed discharged. Similar graphical illustration under lognormal distribution is shown in Fig.5b. The predicted discharges were 431,439 and 443 m3/s respectively while the R2 for the predicted discharges was 0.9790. The forecasted discharges are significantly higher and do not compare well with the observed discharges. Figure 5c shows the graphical illustration for log-Pearson type (iii) distribution. The predicted discharges were 425, 432 and 436m3/s respectively at the same return periods and the coefficients of determination R2 0.9909. The normal distribution gave the best fit and minimum absolute differences between the predicted and observed discharges and is therefore selected as the best distribution using this plotting position.

4.6 Cunnane’s Plotting Position The observed discharges in Figs 6a, b and c were 410, 419 and 423m3/s at return periods of 25, 50 and 100 years respectively. Figure 6a shows the graphical illustration of the annual maximum discharges against the non-exceedence probability using the normal distribution. At the return periods of 25, 50 and 100 years, the predicted discharges were 412, 418 and 421m3/s respectively and coefficient of determination R2 was 0.9921. Figure 6b shows the graph similar graph for lognormal distribution. The predicted discharges at return periods of 25, 50 and 100 years were 435, 443 and 447m3/s respectively and R2 for the predicted discharges was 0.9784. Figure 6c shows graphical illustration for log-Pearson type (iii) distribution. The predicted discharges at the same return periods were 425, 432 and 435m3/s respectively and R2 was 0.9906. The observed and predicted discharges in Fig 6a compare favourably having the least absolute differences unlike Fig.6b where the observed and predicted discharges were significantly higher. In the light of this, normal distribution is taken as the best under this plotting position

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y = 201.67e0.0074x

R2 = 0.9908

y = 150.4e0.0103x

R2 = 0.9761

100

150

200

250

300

350

400

450

500

0 10 20 30 40 50 60 70 80 90 100

Probability of non-exedence(%) (a)

Ann

ual m

axim

um d

isch

arge

s (m

3 /s) Observed

Predicted

y = 150.4e0.0103x

R2 = 0.9761

y = 188.1e0.0087x

R2 = 0.9752

100

150

200

250

300

350

400

450

500

550

0 10 20 30 40 50 60 70 80 90 100

Probaility of non-exceedence (%) (b)

Ann

ual m

axim

um d

isch

arge

s (m

3 /s)

ObservedPredicted

y = 196.51e0.0081x

R2 = 0.989

y = 150.4e0.0103x

R2 = 0.9761

100

150

200

250

300

350

400

450

500

0 10 20 30 40 50 60 70 80 90 100

Probabulity of non-exceedence (%) (c)

Ann

ual m

axim

um d

isch

arge

s (m

3 /s) Observed

Predicted

Figure 3. Normal (a), Lognormal (b) and Log-Pearson type (iii) (c), distributions using Hazen’s plotting position

y = 204.48e0.0072x

R2 = 0.995

y = 146.36e0.0109x

R2 = 0.9761

100

150

200

250

300

350

400

450

0 10 20 30 40 50 60 70 80 90 100

Probability of non-exceedence (%) (a)

Ann

ual m

axim

um d

isch

arge

s (m

3 /s)

ObservedPredicted

y = 191.07e0.0084x

R2 = 0.986

y = 146.36e0.0109x

R2 = 0.9761

100

150

200

250

300

350

400

450

500

10 20 30 40 50 60 70 80 90 100

Probability of non-exceedece (%) (b)

Ann

ual m

axim

um d

isch

arge

s (m

3 /s)

ObservedPredicted

y = 193.46e0.0084x

R2 = 0.9942

y = 146.36e0.0109x

R2 = 0.9761

100

150

200

250

300

350

400

450

500

10 20 30 40 50 60 70 80 90 100

Probability of non-exceedence (%) (c)

Ann

ual m

axim

um d

isch

arge

s (m

3 /s) Observed

Predicted

Figure 4. Normal (a), Lognormal (b) and Log-Pearson type (iii) (c) distributions using Weibull’s plotting position

13


y = 202.83e0.0074x

R2 = 0.9924

y = 149.38e0.0105x

R2 = 0.9761

100

150

200

250

300

350

400

450

10 20 30 40 50 60 70 80 90 100

Probability of non-exceedence (%) (a)

Ann

ual m

axim

um d

isch

arge

s (m

3 /s)

ObservedPredicted

y = 149.38e0.0105x

R2 = 0.9761

y = 189e0.0086x

R2 = 0.979

100

150

200

250

300

350

400

450

500

10 20 30 40 50 60 70 80 90 100

Probability of non-exceedence (%) (b)

Ann

ual m

axim

um d

isch

arge

s (m

3 /s)

ObservedPredicted

y = 149.38e0.0105x

R2 = 0.9761

y = 197.23e0.008x

R2 = 0.9909

100

150

200

250

300

350

400

450

500

10 20 30 40 50 60 70 80 90 100


Ann

ual m

axim

um d

isch

arge

s (m

3 /s) Observed

Predicted

y = 202.23e0.0074x

R2 = 0.9921

y = 149.58e0.0105x

R2 = 0.9761

100

150

200

250

300

350

400

450

500

10 20 30 40 50 60 70 80 90 100

Probaility of non-exceedence (%)(a)

Ann

ual m

axim

um d

isch

arge

s (m

3 /s) ObservedPredicted

y = 188.83e0.0087x

R2 = 0.9784

y = 149.58e0.0105x

R2 = 0.9761

100

150

200

250

300

350

400

450

500

550

10 20 30 40 50 60 70 80 90 100

Probability of non-exceedece (%) (b)

Ann

ual m

axim

um d

isch

arge

s (m

3 /s)

ObservedPredicted

y = 197.09e0.008x

R2 = 0.9906

y = 149.58e0.0105x

R2 = 0.9761

100

150

200

250

300

350

400

450

500

10 20 30 40 50 60 70 80 90 100


Ann

ual m

axim

um d

isch

arge

s (m

3 /s) Observed

Predicted

Figure 5. Normal (a) Lognormal (b) and Log-Pearson type (iii) (c) distributions using Blom’s plotting position

Figure 6. Normal (a), Lognormal (b) and Log-Pearson type (iii) (c) distributions using Cunnane’s plotting position

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4.7 California Plotting Position The observed discharges at return periods of 25, 50 and 100 years were 415,424 and 429 m3/s respectively (see Figs 7a, b and c). Figure 7a shows the graphical illustration of the annual maximum discharges against the probabilities of non-exceedence for the normal distribution using California plotting position. The predicted discharges at return periods of 25, 50 and 100 years were 407, 413 and 416m3/s respectively and coefficient of determination R2 was 0.9939. Although of smaller magnitudes, the predicted discharges compares well observed discharges Also, the minimum root mean square error (RMSE) of 29.06m3/s was estimated under this distribution (see Table 3). Figure 7b shows the similar graph for lognormal distribution. At return periods of 25, 50, and 100 years, the predicted discharges were 425, 432 and 435m3/s respectively and R2 was 0.9887. Figure7c gives the same graphical illustration for log–Pearson type (iii) distribution. At the return periods of 25, 50 and 100 years, the predicted discharges were 422, 429 and 432m3/s respectively and R2 was 0.9946. The log-Pearson type (iii) distribution had the highest fit and the minimum absolute differences between the observed and predicted discharges and is selected as the best using this plotting position. 4.8 Gringorton’s Plotting Position The observed discharges at return periods of 25, 50 and 100 years were 415,424 and 429 m3/s respectively as shown in Figs. 8a, b and c. Figure 8a gives the graphical illustration of the normal distribution using Gringorton’s plotting position. At return periods of 25, 50 and 100 years, the predicted discharges were 411, 417 and 420m3/s respectively and coefficient of determination R2 was 0.9916.These predicted and observed discharges compares well and gave the minimum absolute differences when compared with all other plotting positions used in this statistical analysis. The root mean square error (RMSE) was 35.57m3/s (see Table 3). Figure 8b shows similar graph for lognormal distribution. The predicted discharges at the same return periods were 435, 442 and 446m3/s respectively and R2 was 0.9772. Figure 8c gives graphical illustration for log Pearson type (iii) distribution. The predicted discharges were 424, 431 and 435m3/s respectively at the same return periods and R2 for the predicted discharges was 0.9900. The regression coefficient coupled with the minimum absolute difference show that normal distribution gave the best fit to the annual peak flow using this plotting position. 4.9 Chegodajev’s Plotting Position The observed discharges at return periods of 25, 50 and 100 years were 412, 420 and 425 m3/s respectively as shown in Figs. 8a, b and c. Figure 9a shows the normal distribution of the annual maximum discharges against the. non-exceedence probabilities using Chegodajev’s plotting position. At return periods of 25, 50 and 100 years, the predicted discharges were 409, 415 and 418m3/s respectively and coefficient of determination R2 was 0.9931. The root mean square error (RMSE) was 33.32m3/s (see Table 3). Similar graphical illustration for lognormal distribution is shown in Fig.9b. At return periods of 25, 50, and 100

15


years the predicted discharges were 433, 440 and 444m3/s respectively and R2 was 0.9809. Figure 9c shows the graphical illustration for log-Pearson type (iii) distribution. The predicted discharges were 426,432 and 436m3/s at return periods of 25, 50 and 100 years respectively. The coefficient R2 was 0.9918. Minimum absolute differences were obtained between the predicted and observed discharges in Fig. 9a and the highest coefficient of determination was obtained under normal distribution. Table 3 shows the coefficients of determination, root mean square errors (RMSE) and absolute differences between observed and predicted discharges. The best distribution was determined by considering the absolute differences between the observed and the predicted discharges, coefficients R2 of their regression equations and the root mean square errors (Kottegodal, 1980; Benjamin, Garry and Jean, 2001). The minimum absolute differences between the predicted and observed discharges under each plotting position were obtained using normal distribution. However, the Weibull’s plotting position that has the highest coefficient of determination R2of 0.9950 under normal distribution had higher absolute differences when compared with those of Cunnane and Gringorton under the same normal distribution. California plotting position had the minimum (RMSE) under the normal distribution. As indicated earlier, the best formula is selected based on purpose of investigation and the distribution under investigation. The Weibull’s plotting position had the highest R2 and is hereby selected as the best under normal distribution. This is in agreement with Alatise, (1998) and Chen, et al, (2002) where the same indicators were used in selecting optimum formulae in flood analyses. Similarly, log-Pearson type (iii) and normal distributions had the highest coefficient of determination R2 using the California formula. Generally, the performances of the probability distributions were satisfactory as none of them had coefficient of determination R2 that was less than 0.98. For the purpose of accurate flood forecast which is instrumental to decisions making on flood mitigation measures, graphical approaches are good and is encouraged.

16


y = 202.06e0.0073x

R2 = 0.9939

y = 154.55e0.0103x

R2 = 0.9761

100

150

200

250

300

350

400

450

0 10 20 30 40 50 60 70 80 90 100

Probability of non-exceedence (%)(a)

Ann

ual m

axim

um d

isch

arge

s (m

3 /s)

ObservedPredicted

y = 191.35e0.0083x

R2 = 0.9887

y = 154.55e0.0103x

R2 = 0.9761

100

150

200

250

300

350

400

450

500

0 10 20 30 40 50 60 70 80 90 100


Ann

ual m

axim

um d

isch

arge

s (m

3 /s)

ObservedPredicted

y = 197.79e0.0079x

R2 = 0.9946

y = 154.55e0.0103x

R2 = 0.9761

100

150

200

250

300

350

400

450

500

0 10 20 30 40 50 60 70 80 90 100


Ann

ual m

axim

um d

isch

garg

es (m

3 /s)

ObservedPredicted

Figure 7. Normal (a), Lognormal (b) and (c) Log-Pearson type (iii) (c) distributions using California plotting position

y = 202.01e0.0074x

R2 = 0.9916

y = 149.91e0.0104x

R2 = 0.9761

100

150

200

250

300

350

400

450

500

10 20 30 40 50 60 70 80 90 100

Probability of non-exceedece (%) (a)

Ann

ual m

axim

um d

isch

garg

es (m

3 /s) Observed

Predicted

y = 188.55e0.0087x

R2 = 0.9772

y = 149.91e0.0104x

R2 = 0.9761

100

150

200

250

300

350

400

450

500

0 10 20 30 40 50 60 70 80 90 100


Ann

ual m

axim

um d

ischa

rges

(m3 /s)

ObservedPredicted

y = 196.87e0.008x

R2 = 0.99

y = 149.91e0.0104x

R2 = 0.9761

100

150

200

250

300

350

400

450

500

10 20 30 40 50 60 70 80 90 100


Ann

ual m

axim

um d

isch

arge

s (m

3 /s) Observed

Predicted

Figure 8. Normal (a), Lognormal (b) and (c) Log-Pearson type (iii) (c) distributions using Gringorton’s plotting position

17


y = 202.73e0.0073x

R2 = 0.9931

y = 148.77e0.0106x

R2 = 0.9761

100

150

200

250

300

350

400

450

10 20 30 40 50 60 70 80 90 100Probability of non-exceedence (%)

(a)

Ann

ual m

axim

um d

isch

arge

s (m

3 /s)

ObservedPredicted

y = 189.48e0.0086x

R2 = 0.9809

y = 148.77e0.0106x

R2 = 0.9761

100

150

200

250

300

350

400

450

500

10 20 30 40 50 60 70 80 90 100


Ann

ual M

axim

um D

isch

arge

s (m

3 /s)

ObservedPredicted

y = 197.62e0.008x

R2 = 0.9918

y = 148.77e0.0106x

R2 = 0.9761

100

150

200

250

300

350

400

450

500

0 10 20 30 40 50 60 70 80 90 100


Ann

ual m

axim

um d

iscg

arge

s (m

3 /s)

ObservedPredicted

Figure 9. Normal (a), Lognormal (b) and (c) Log-Pearson type (iii) (c) distributions using Chegodajev’s

plotting position

18


Table 3: Coefficients of determination, root mean square errors (RMSE) and absolute differences between observed and predicted discharges

Plotting Positions

Probability Distributions

Log-Pearson Normal Lognormal type (iii)

Correlation coefficients

R2

Root Mean Square Errors

(RMSE) (m3/s)

Return Periods (years)

Absolute differences

(m3/s)

Hazen

Weibull Blom

Cunnane California Gringorton Chegodajev

0.9908 0.9950 0.9924 0.9921 0.9939 0.9916 0.9931

0.9752 0.9860 0.9790 0.9784 0.9887 0.9772 0.9809

0.9890 0.9942 0.9909 0.9906 0.9946 0.9900 0.9918

Hazen

Weibull Blom


35.84 35.09 35.75 35.45 29.06 35.57 33.32

43.09 36.59 40.86 41.26 31.87 41.95 37.73

40.24 39.25 39.12 39.32 31.89 39.67 36.60

25 50 100 25 50 100 25 50 100

Hazen

Weibull Blom


06 09 04 02 08 04 03

04 12 03 01 11 02 05

03 14 05 02 13 00 07

30 11 22 25 10 28 21

29 09 17 24 08 27 20

28 08 16 21 06 26 19

23 16 16 15 07 17 14

23 14 10 13 05 16 10

21 13 09 12 03 15 11

19


4. CONCLUSION In this study, the peak discharges were plotted against their hydrologic years. Three probability distributions and seven plotting positions were fitted to the annual maximum discharges of River Osun at Apoje gauging station. The performances of the probability distributions were assessed using the coefficients of determination, root mean square errors (RMSE) and absolute differences between predicted and observed discharges. The following conclusions were drawn from the study:

• The annual maximum discharges of River Osun at Apoje gauging station vary in magnitude and space, ranging from 169 to 400m3/s within between 1982 and 1999. The minimum “peak flow” in the dry season was 72m3/s. The minimum and maximum mean of the annual flows were 82 and 189m3/s respectively.

• The normal distribution had the highest coefficient of determination using Welbull’s plotting position while lognormal and log-Pearson type (iii) distributions had the highest fit when matched with California plotting positions.

• Normal distribution had minimum (RMSE) when matched with California and Weibull’s plotting positions. The lognormal distribution had the minimum (RMSE) when matched with California plotting position.

• The minimum absolute differences at return periods of 25, 50 and 100 years were obtained under the normal distribution when matched with Cunnane plotting position.

The flood frequency analysis shows that under the normal distribution, the Weibull’s formula gives the best fit while California formula gives the best fit under the log-Pearson type (iii) distribution and can therefore be proposed for the river basins in the rain forest belt. Generally, Apoje sub-basin has abundant water potential and if well harnessed can be used for various purposes in the basin.

5. REFERENCES

Adeboye O. B. 2005. Flood Characteristics and Potential Reservoir Capacity of River Osun at Apoje Gauging Station, M. Eng Thesis, Unpublished, Department of Agricultural Engineering, Federal University of Technology, Akure, Nigeria, 110pp

Akso, H. 2000. Use of Gamma Distribution in Hydrological Analysis, Turks Journal of Engineering and Environmental Sciences, 24:419-428

Alatise, M. O. 1998. Comparison of the Log-Normal and Pearson type (iii) Methods in Analyzing the Flood Characteristics of River Owena in Ondo State, N.S.E Technical Transactions, 33(4), 18 pp

Benjamin, F. P., D. T. Gary, and C. R. Jeane. 2001. Estimating the Magnitudes and Frequency of Floods in the Rural Basin of the North Carolina-Revised, U.S.G.S, Water Resources Investigation Reports 01-4207, Raleigh, North Carolina, 49pp

Blom, G. 1958. Statistical Estimates and Transformed Beta-variable, John Willey & Sons, N.Y Chegodajev, N. N. 1955. Formulas for Calculation of the Confidence of Hydrologic Quantities,

Handbook of Applied Hydrology, V.T. Chow, ed, McGraw Hill, N.Y

20


Chen.Y. U. Y., V. G. Pieter, and Z. Sha, 2002, A Study of the Parameter Estimation Methods for Pearson Type (iii) Distributions in Flood Frequency Analysis, IASH Publications, No 271, Iceland, 350-407.

Cunnane, C. 1978. Unbiased Plotting Position- A review, Journal of Hydrology, 37:205-222 Flows in California stream, 1923. Bulletin Number 5, Division of Engineering and Irrigation, California,

Department of Public works, California state Printing office, Sacramento, California Chow, V. T., R. D. Maidment, and W. M. Larry, 1988. Applied Hydrology, International edition,

McGraw Hill Inc, Singapore, 350-407 Claps, P. and F. Laio. 2003. Can Continuous Streamflow Data Support Frequency Analysis? An

Alternatives to the Partial Duration Series Approach, Water Resources Research, 39 (8), 1-11 Ekanayake. S. T., J. F. Cruise, 1993. Comparisons of Weibull and exponential based partial duration

stochastic flood models, Stochastic Hydrology and Hydraulics, 7: 283-297 Federal Republic of Nigeria. 1982. Semi Detailed Soil Survey of Osun, Ona and Ogun River Basins,

Prepared by Department of Soil and Science, Obafemi Awolowo University, Ile-Ife, Nigeria. Gringorton, I. I. 1963. A Plotting Position for Extreme Probability Paper, Journal of Geophysics

Research, 68(3), 813-814 Green I. R. A., and D. Stephenson. 1985. Comparison of Urban Drainage Models for use in South Africa,

WRC Report No 115/6/86., Water Resources commission, Pretoria RSA. Hazen, A. 1914. Storage to be Provided in Impounding Reservoir for Municipal Water Supply,

Transaction of ASCE, 77:1547-1550 Hebson, C. S and C. Cunnane. 1987. Assessments of the Site and Regional Flood Frequency Estimation,

In VP Singh(ed), Hydrology frequency modelling, Reidel Norwell, Massachusetts, U.S.A, 433-448 Hirsch, R. M. and J. R. Stedinger, 1987. Plotting Positions for Historic Flood and their Precision, Water

Resources Research, 23(4), 715-727 Hoskin J. R. M and J. R. Wallis. 1997. Regional Frequency Analysis: An approach based on L-Moments,

Cambridge University press, UK, 22pp Hromadka, 11 T. V and R. J. Whitley, 1989. Checking Flood Frequency Curves using Rainfall Data,

Journal of Hydraulic Engineering, ASCE, 115(4), N.Y, 544-548 Kite, G. M. 1977. Frequency and Risk Analysis in Hydrology, Water Resources Publications, Fort

Collins, N.Y Kjeldsen, T. R., J. C. Smithers and R. E. Schulze, 2001. Flood Frequency Analysis at ungauged sites in

the KwaZulu-Nata Province, South Africa, Water SA, 27(3), 315-324 Kottegodal, N. T. 1980. Stochastic Water Resources Technology, 1st edition, Macmillan Press Ltd,

London, 208-215 Moughamiam, M. S., D. B. Mclaughlin, and R. E., Bras. 1987. Estimation of Flood Frequency: An

Evaluation of two Derived Distribution Procedures, Water Resources Research, 23(7), 1309-1319 O’ Donnell. T. 1985. A Direct three-parameter Muskingum procedure incorporating lateral inflow,

Journal of Hydrological Sciences, 30(4), 494-495 Robert, M. H. 1987. Plotting Positions for Historic Floods and their Precision, American Geophysical

Union, 23(4), Washington, D.C, U.S.A, 715-719 Sumioka, S. S., D. L Kresh, and K. D Kasnick. 1997. Magnitudes and Frequencies of Flood in

Washington, US Geological Survey, Water Resources Investigation Report, 97-4277 W. A, 15 pp

21


Tadesse N. 2006. Surface Water Potentials of the Hantebet Basin, Tigray, Northern Ethiopia”, Agricultural Engineering International: CIGR Ejournal, Vol. VIII.

Tewolde. M. H. and J. C. Smitthers. 2006. Flood Routing in Ungauged Catchments using Muskingum methods, WRC, Water S.A 32(3), RSA

Tilahun K. 2006. The Characterization of Rainfall in the arid and semi arid regions of Ethiopia, Water SA, 32(3), 429-439, http://www.wrc.org.za

Topaloglu, F. 2002, Determining Suitable Probability Distributions Models for Flow and Precipitation Series of Seyhan River Basin, Turk Journal Agriculture, 26, 187-194

Van Bladeren, D. 1993. Application of Historic data in Flood Frequency Analysis for Natal and Transkei Regions. In proceeding of the 6th South African National Hydrologic Symposium Vol.1, Pietermaritzburg, RSA

Van Thanh-Van-Nguyen, V. N., I. Nophadel., B. Bernarb. 1989. New Plotting Position Formulae for Pearson Type (iii) distribution, Journal of Hydraulic Engineering, Transaction of ASCE, 115(6), N.Y, 709–730.

Viessman, W. J and L. Lewis. 1996. Introduction to Hydrology, 4th edition, HapercollinCollege Publisher, N.Y, 760pp

Weibull, W. 1939. A Statistical Theory of Strength of Materials Ing. Vet. A.K, Handl., 151, Genelstabens Litografiska Anstals Forlg Stocklholm, Sweden

U.S. Water Resources Council, 1981, Guidelines for Determining Flood Flow Frequency, Bulletin 17A, U.S Geological Survey, Washington, D.C

Documents

Performance of Probability Distributions and Plotting Positions in